The Gamma function for positive arguments can be defined with the integral
$$ \Gamma(\alpha) = \int_0^\infty x^{\alpha-1} e^{-x}\,dx $$
The function $ x^{\alpha-1} e^{-x} $ is called the Gamma distribution when normalized.
How are the two names related?
- Did the Gamma function get this name and the $\Gamma$ symbol because the distribution was already called Gamma distribution?
- Or backwards, did the Gamma distribution get named from the Gamma function?
- Or were the two named together at the same time?
- Or are the two names independent and only accidentally the same letter, despite the apparent connection?
This question came up in a discussion about spelling mathematical terms, and while it's probably not actually relevant for that, I'm now curious about the origin of the names.